Notation in probability and statistics
Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols.
Probability theory
 Random variables are usually written in upper case roman letters: X, Y, etc.
 Particular realizations of a random variable are written in corresponding lower case letters. For example, x_{1}, x_{2}, …, x_{n} could be a sample corresponding to the random variable X. A cumulative probability is formally written $P(X\leq x)$ to differentiate the random variable from its realization.
 The probability is sometimes written $\mathbb {P}$ to distinguish it from other functions and measure P so as to avoid having to define " P is a probability" and $\mathbb {P} (A)$ is short for $P(\{\omega :X(\omega )\in A\})$, where $\omega$ is an event and $X(\omega )$ a corresponding random variable.
 $\mathbb {P} (A\cap B)$ or $\mathbb {P} [A\cap B]$ indicates the probability that events A and B both occur.
 $\mathbb {P} (A\cup B)$ or $\mathbb {P} [A\cup B]$ indicates the probability of either event A or event B occurring ("or" in this case means one or the other or both).
 σalgebras are usually written with upper case calligraphic (e.g. ${\mathcal {F}}$ for the set of sets on which we define the probability P)
 Probability density functions (pdfs) and probability mass functions are denoted by lower case letters, e.g. f(x).
 Cumulative distribution functions (cdfs) are denoted by upper case letters, e.g. F(x).
 Survival functions or complementary cumulative distribution functions are often denoted by placing an overbar over the symbol for the cumulative:${\overline {F}}(x)=1F(x)$
 In particular, the pdf of the standard normal distribution is denoted by φ(z), and its cdf by Φ(z).
 Some common operators:

 X is independent of Y is often written $X\perp Y$ or $X\perp \!\!\!\perp Y$, and X is independent of Y given W is often written
 $X\perp \!\!\!\perp Y\,\,W$ or
 $X\perp Y\,\,W$
 $\textstyle P(A\mid B)$, the posterior probability, is the probability of $\textstyle A$ given $\textstyle B$, i.e., $\textstyle A$ after $\textstyle B$ is observed.
Statistics
 Greek letters (e.g. θ, β) are commonly used to denote unknown parameters (population parameters).
 A tilde (~) denotes "has the probability distribution of".
 Placing a hat, or caret, over a true parameter denotes an estimator of it, e.g., ${\widehat {\theta }}$ is an estimator for $\theta$.
 The arithmetic mean of a series of values x_{1}, x_{2}, ..., x_{n} is often denoted by placing an "overbar" over the symbol, e.g. ${\bar {x}}$, pronounced "x bar".
 Some commonly used symbols for sample statistics are given below:
 Some commonly used symbols for population parameters are given below:
 the population mean μ,
 the population variance σ^{2},
 the population standard deviation σ,
 the population correlation ρ,
 the population cumulants κ_{r},
 $x_{(k)}$ is used for the $k^{th}$ order statistic, where $x_{(1)}$ is the sample minimum and $x_{(n)}$ is the sample maximum from a total sample size n.
Critical values
The αlevel upper critical value of a probability distribution is the value exceeded with probability α, that is, the value x_{α} such that F(x_{α}) = 1 − α where F is the cumulative distribution function. There are standard notations for the upper critical values of some commonly used distributions in statistics:
Linear algebra
 Matrices are usually denoted by boldface capital letters, e.g. A.
 Column vectors are usually denoted by boldface lower case letters, e.g. x.
 The transpose operator is denoted by either a superscript T (e.g. A^{T}) or a prime symbol (e.g. A′).
 A row vector is written as the transpose of a column vector, e.g. x^{T} or x′.
Abbreviations
Common abbreviations include:
See also
References
 Halperin, Max; Hartley, H. O.; Hoel, P. G. (1965), "Recommended Standards for Statistical Symbols and Notation. COPSS Committee on Symbols and Notation", The American Statistician, 19 (3): 12–14, doi:10.2307/2681417, JSTOR 2681417
External links
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